Random $K$-satisfiability ($K$-SAT) is a model system for studying typical-case complexity of combinatorial optimization. Recent theoretical and simulation work revealed that the solution space of a random $K$-SAT formula has very rich structures, including the emergence of solution communities within single solution clusters. In this paper we investigate the influence of the solution space landscape to a simple stochastic local search process {\tt SEQSAT}, which satisfies a $K$-SAT formula in a sequential manner. Before satisfying each newly added clause, {\tt SEQSAT} walk randomly by single-spin flips in a solution cluster of the old subformula. This search process is efficient when the constraint density $α$ of the satisfied subformula is less than certain value $α_{cm}$; however it slows down considerably as $α> α_{cm}$ and finally reaches a jammed state at $α\approx α_{j}$. The glassy dynamical behavior of {\tt SEQSAT} for $α\geq α_{cm}$ probably is due to the entropic trapping of various communities in the solution cluster of the satisfied subformula. For random 3-SAT, the jamming transition point $α_j$ is larger than the solution space clustering transition point $α_d$, and its value can be predicted by a long-range frustration mean-field theory. For random $K$-SAT with $K\geq 4$, however, our simulation results indicate that $α_j = α_d$. The relevance of this work for understanding the dynamic properties of glassy systems is also discussed.
We present the study of the landscape structure of athermal soft spheres both as a function of the packing fraction and of the energy. We find that, on approaching the jamming transition, the number of different configurations available to the system has a steep increase and that a hierarchical organization of the landscape emerges. We use the knowledge of the structure of the landscape to predict the values of thermodynamic observables on the edge of the transition.
Random constraint satisfaction problems (CSP) have been studied extensively using statistical physics techniques. They provide a benchmark to study average case scenarios instead of the worst case one. The interplay between statistical physics of disordered systems and computer science has brought new light into the realm of computational complexity theory, by introducing the notion of clustering of solutions, related to replica symmetry breaking. However, the class of problems in which clustering has been studied often involve discrete degrees of freedom: standard random CSPs are random K-SAT…
Critical jamming transitions are characterized by an astonishing degree of universality. Analytic and numerical evidence points to the existence of a large universality class that encompasses finite and infinite dimensional spheres and continuous constraint satisfaction problems (CCSP) such as the non-convex perceptron and related models. In this paper we investigate multilayer neural networks (MLNN) learning random associations as models for CCSP which could potentially define different jamming universality classes. As opposed to simple perceptrons and infinite dimensional spheres, which are …
We present a parallel derivation of the Thouless-Anderson-Palmer (TAP) equations and of an effective potential for the negative perceptron and soft sphere models in high dimension. Both models are continuous constrained satisfaction problems with a critical jamming transition characterized by the same exponents. Our analysis reveals that a power expansion of the potential up to the second order represents a successful framework to approach the jamming line from the SAT phase (the region of the phase diagram where at least one configuration verifies all the constraints), where the ground-state …
We discuss a resource-competition model, which takes the MacArthur's model as a platform, to unveil interesting connections with glassy features and jamming in high dimension. This model presents two qualitatively different phases: a "shielded" phase, where a collective and self-sustained behavior emerges, and a "vulnerable" phase, where a small perturbation can destabilize the system and contribute to population extinction. We first present our perspective based on a strong similarity with continuous constraint satisfaction problems in their convex regime. Then, we discuss the stability in te…
In this letter we study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, which takes advantage of a mapping to an equilibrium disordered system, proves that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we un…
